arXiv subjects
Publications and source records attributed to Korn'el Szlach'anyi.
We study Frobenius 1-morphisms $ı$ in an additive bicategory $\c$ satisfying the depth 2 condition. We show that the 2-endomorphism rings $\c^2(ı\x\ib,ı\x\ib)$ and $\c^2(\ib\xı,\ib\xı)$ can be equipped with dual Hopf algebroid structures. We prove also that a Hopf algebroid appears as the solution of the above abstract symmetry problem if and only if it possesses a two sided non-degenerate integral.
Motivated by the study of depth 2 Frobenius extensions we introduce a new notion of Hopf algebroid. It is a 2-sided bialgebroid with a bijective antipode which connects the two, left and right handed, structures. While all the interesting examples of the Hopf algebroid of J.H. Lu turn out to be Hopf algebroids in the sense of this paper, there exist simple examples showing that our definition is not a special case of Lu's. Our Hopf algebroids, however, belong to the class of $\times_L$-Hopf algebras proposed by P. Schauenburg. After discussing the axioms and some examples we study the theory of non-degenerate integrals in order to obtain duals of Hopf algebroids.